The basic unit of information processing in modern day computers is a bit, which can exist in one of two states: 0 or 1. In quantum computing, the basic unit of information processing is a qubit. Like a bit, the qubit can also exist in two states, which states are denoted as state |0 and state |1. But unlike a bit, a qubit can exist in superposition states, which are a linear combination of state |0 and state |1. For example, if the qubit were to be denoted by |ψin, then the superposition states of the qubit can be described by the expression: |ψin=α|0+β|1, where α and β are complex amplitudes (also known as probability amplitudes), and where α2+β2=1. Therefore, the qubit |ψin can exist in any of the states described by variables α and β. This property of qubits is essential in quantum computing. Therefore, if the qubit |ψin is prepared in a superposition state for quantum computing, then it is essential that this superposition state be maintained throughout the computing process, meaning that the state may need to be recovered after it is determined (i.e., read).
Results of a quantum computation require that one or more qubits of the quantum computer be measured. Two kinds of measurements can be carried out on a qubit |ψin: a strong measurement and a weak measurement. In a strong measurement, the qubit ceases to exist in the superposition state, and collapses into one of its so-called eigenstates, i.e., |0 and |1. An example of a strong measurement is a detector detecting a click in a cavity quantum electrodynamics (QED) system, where measuring a click indicates that the qubit |ψin has collapsed into state |1. Because of the collapse of the superposition state, information related to the original state of the system cannot be recovered. In other words, the values of α and β are lost.
In a weak measurement, full collapse into the eigenstate does not take place, and it is possible to reverse the measurement so that the original state of the system is recovered. This reversal is possible, because for weak measurements, information of the probability amplitudes α and β involved in the superposition of the states is retained. One example of a weak measurement is leakage of a field inside a cavity QED system. For example, if a detector does not register a click after a time τ, then the state evolves into |ψ(τ)d=(α|0+e−Γτβ|1)/√{square root over (|α|2+|β|2e−2Γτ)}, where Γ is the cavity decay rate, and τ is the duration of the weak measurement.
A weak measurement thus retains partial information on the state of the qubit |ψin: information regarding α and β are present, although the amplitude of the state is dampened. Thus, unlike strong measurements, weak measurements provide the possibility of recovering the qubit to its original superposition state. This also applies for weak measurements implemented in systems other than cavity QED systems.
U.S. Pat. No. 8,350,587 describes methods and systems for recovering a state of a qubit to its original state where the qubit has been transformed by a weak measurement and hence amplitude dampened. The '587 patent is incorporated herein in its entirety, and because of its general relevance to the inventive techniques that follow, some time is spent in this Background discussing the technique of the '587 patent.
As shown in FIG. 1, the system 100 disclosed in the '587 patent uses an additional qubit, referred to as an ancillary qubit 103, to recover the measured qubit |ψin to its original state after a weak measurement 101. System 100 includes a reversing circuit 102, which receives the dampened amplitude |ψd of the weakly-measured qubit |ψin as one of its inputs. Reversing circuit 102 also receives the ancillary qubit 103 (|0). The state of the ancillary bit 103 at the output of the reversing circuit 102 is measured by a detector 104.
The output of the detector 104 is fed to a controller 105, which, based on the detector's 104 output, determines whether the state of the qubit |ψ has been recovered. If the controller 105 determines that the state has not been recovered, then the system performs another iteration: another ancillary bit 103 is input to the reversing circuit 102; the qubit |ψout at the output of the reversing circuit 102 is fed back to the reversing circuit; and a rotation angle of Hadamard gate 106 is appropriately modified, as exemplified by the dotted lines. This process is iteratively repeated until the state of the qubit |ψout is determined to be recovered.
As discussed above, the original state of the qubit |ψin can be expressed as α|0+β|1. Due to the weak measurement, the state of the qubit |ψd evolves into: |ψd=(α|0+e−Γτβ|1)/√{square root over (|α|2+|β|2e−2Γτ)}. For simplicity, we denote N0=√{square root over (|α|2+|β|2e−2Γτ)}, and rewrite the state of qubit v) after a weak measurement as:|ψd=(α|0+e−Γτβ|1)/N0  (B)
The reversing circuit 102 includes a Hadamard gate Hθ 106 and a CNOT gate 107. The two-qubit input to the reversing circuit 102 (i.e., |ψd and |0) can be represented as |ψ12, and the state of this two-qubit system can be expressed as:|ψ1=(α|0+e−Γτβ|1)/N0|0  (C)The Hadamard gate Hθ 106 transforms the ancillary qubit 103 (|0) into cos θ|0+sin θ|1, where θ is a rotation angle. Because this is the first iteration of the system, θ is denoted as θ1. Thus, the result of the Hadamard gate Hθ 106 transforming qubit |ψ1 can be expressed as:|ψ2=(α|0+e−Γτβ|1)/N0(cos θ1|0+sin θ1|1)  (D)|ψ2=(α cos θ1|0|0+e−Γτβ cos θ1|1|0+α sin θ1|0|1+e−βτ sin θ1|1|1)/N0  (E)
State |ψ2, as expressed in Equation (E), is then input to the CNOT gate 107. As is well known, the CNOT gate 107 is a controlled NOT gate having a control input (C) and a target input (T). The CNOT gate flips the state of the target input only if the control input is |1. In Equation (E), for each of the four terms that are summed, the first qubit state corresponds to the control input and the second qubit state corresponds to the target input. Thus for each of the terms in Equation (E) where the first qubit is |1, the second qubit will be flipped. Thus, the CNOT gate 104 transforms the state |ψ2 as follows:|ψ3=(α cos θ1|0|0+e−Γτβ cos θ1|1|1+α sin θ1|0|1+e−Γτβ sin θ1|1|0)/N0  (F)
Rearranging the terms in Equation (F) results in:|ψ3=((α cos θ1|0+e−Γτβ cos θ1|1)|0+(α sin θ1|0+e−Γτβ sin θ1|1)1)/N0  (G)In each of the two summed terms of Equation (G), the right side term in the tensor product represents the state of the ancillary bit while the left hand term represents the state of the system after CNOT transformation. Therefore, if the state of the ancillary bit were to be measured, the result of the measurement would provide the state of the qubit |ψout. For example, if state of the ancillary bit measured by the detector 104 is |0, then the state of the qubit |ψout would be α cos θ1|0+e−Γτβ sin θ1|1; and if the measured state is |1, then the state of the qubit |ψout would be (α sin θ1|0+e−Γτβ cos θ1|1)/N0.
Assume θ1 had earlier been set by the controller 105 to tan−1 eΓτ. Using this value for θ1 in Equation (G), if the state of the ancillary qubit is measured to be |0, then the state of the qubit |ψout would be α|0+β|1. As previously stated, this is the original state of the qubit |ψin, which is thus recovered to its original state, despite the amplitude dampening caused by the weak measurement.
If the detector 104 measures a |1, then the state of the qubit |ψout is:
                                          ❘            ψ                    〉                =                              1                          N              1                                ⁢                      (                                          α                ⁢                                                    0                  〉                                            +                                                ⅇ                                                            -                      2                                        ⁢                                                                                  ⁢                    Γ                    ⁢                                                                                  ⁢                    τ                                                  ⁢                β                ⁢                                                    1                  〉                                                      )                                              (        H        )            where N1=√{square root over (|α|2+|β|2e−4Γτ)}. Thus, the state of the qubit |ψout has not been recovered to the original state |ψin. To try and recover the original state, the controller 105 carries out another iteration of the above described process, but this time sets the value of θ to θ2=tan−1 e2Γτ, where θ2 indicates a second iteration. Additionally, the qubit |ψout is fed back to the reversing circuit 102. At the end of the second iteration, the controller 105 again measures the state of the ancillary bit. If the measured state is |0, then the original state of the qubit |ψin would be recovered as |ψout. On the other hand, if the measured state is |1, then another iteration would be carried out.
In general, the controller 105 can carry out N iterations to recover the original state of the qubit |ψin, with the value of θn in each iteration N set to:θn=tan−1e2N-1Γτ  (I)
While reversing the qubit |ψin using the reversing circuit 102 may require a number of iterations, the time required for these iterations is far less than the time required for a second weak measurement. For example, in a cavity QED system, the time required for a second weak measurement on a qubit can be in the order of a few milliseconds, whereas the time for a single iteration of the reversing circuit 102 can be in the order of tens of microseconds—approximately two orders of magnitude lower. Therefore, even if a few iterations of the reversing circuit 102 may be required to recover the original state of the qubit, the total time is far less than that required for reversing using a second weak measurement.
It is understood that the system 100 of FIG. 1, and in particular the reversing circuit 102, can be implemented in any physical system that can support a quantum computer. Examples of such physical systems are Josephson junctions, optical lattices, quantum dots, nuclear magnetic resonance systems, cavity quantum electrodynamics (QED), etc.
A person skilled in the art will appreciate that high level quantum gates such as CNOT gates can be represented as a combination of universal quantum gates. As such, it is well known that the CNOT gate 107 of reversal circuit 102 can be built using two Hadamard gates 110 and 112 and phase gate 111, as shown in FIG. 2. Such decomposition of the CNOT gate 107 is beneficial when the physical system is a cavity QED system because implementation of Hadamard gates and phase gates in cavity QED systems are well understood in the art.
The just described-technique of the '587 shows utility in providing memory systems employing qubits, as it allows qubits to be read and then effectively recovered on small time scales. However, this technique does not deal with the possibility that two qubits in a memory system may become entangled. This disclosure expands the technique of the '587 to a two-qubit system, and shows that quantum entanglement, an important source of information in quantum computing, can be protected from amplitude dampening, whether such dampening occurs because of use of a weak measurement or otherwise. Protection of quantum entanglement allows the original states of the qubits to be recovered, or partially recovered with a certain probability.